Prove that the numbers from $1$ to $16$ can be written in a line, but cannot be written in a circle so that the sum of any two adjacent numbers is square of a natural number.

There are 300 apples, any two of which differ in weight by no more than twice. Prove that they can be arranged in packages of two apples so that any two packages differ in weight by no more than one and a half times.

On sides $AB$ and $BC$ of an equilateral triangle $ABC$ are taken points $D$ and $K$, and on the side $AC$ , points $E$ and $M$ so that $DA + AE = KC +CM = AB$. Prove that the angle between lines $DM$ and $KE$ is equal to $60^o$.

The company employs 50,000 people. For each of them, the sum of the number of his immediate superiors and his immediate subordinates is equal to 7. On Monday, each employee of the enterprise issues an order and gives a copy of this order to each of his direct subordinates (if there are any). Further, every day an employee takes all the basics he received on the previous day and either distributes them copies to all your direct subordinates, or, if any, he is not there, he carries out orders himself. It turned out that on Friday no papers were transferred to the institution. Prove that the enterprise has at least 97 bosses over whom there are no bosses.

Segments $AB$, $BC$ and $CA$ are, respectively, diagonals of squares $K_1$, $K_2$, $K3$. Prove that if triangle $ABC$ is acute, then it completely covered by squares $K_1$, $K_2$ and $K_3$.

The numbers from 1 to 37 are written in a line so that the sum of any first several numbers is divided by the number following them. What number is worth in third place, if the number 37 is written in the first place, and in the second, 1?

Find all pairs of prime numbers $p$ and $q$ such that $p^3-q^5 = (p+q)^2$.

In Mexico City, to limit traffic flow, each private car is set two days of the week on which it cannot go out onto the city streets. A family needs to have at least 10 cars at its disposal every day. What is the smallest number of cars can get by with the seven if its members can choose forbidden days for your cars?

A regular $1997$-gon is divided into triangles by non-intersecting diagonals. Prove that exactly one of them is acute-angled.

The numbers $1, 2, 3, ..., 1000$ are written on the board. Two people take turns erasing one number at a time. The game ends when two numbers remain on the board. If their sum is divisible by three, then the one who made the first move wins. if not, then his partner. Which one will win if played correctly?

There are 300 apples, any two of which differ in weight by no more than three times. Prove that they can be arranged into bags of four apples each so that any two bags differ in weight by no more than than one and a half times.

Let's call several numbers written in a row a 'combination of numbers'. In the country of Robotland, some combinations of numbers have been declared prohibited. It is known that there are a finite number of forbidden combinations and there is an infinite decimal fraction that does not contain forbidden combinations. Prove that there is an infinite periodic decimal fraction that does not contain prohibited combinations.

Given a set of $1997$ numbers such that if each number in the set, replace with the sum of the rest, you get the same set. Prove that the product of numbers in the set is equal to $0$.

Given triangle $ABC$. Point $B_1$ bisects the length of the broken line $ABC$ (composed of segments $AB$ and $BC$), point $C_1$ bisects the length of the broken line$ACB$, point $A_1$ bisects the length of of the broken line $CAB$. Through points $A_1$, $B_1$ and $C_1$ straight lines $\ell_A$ ,$\ell_B$, $\ell_C$ are drawn parallel to the bisectors angles $BAC$, $ABC$ and $ACB$ respectively. Prove that the lines $\ell_A$, $\ell_B$ and $\ell_C$ intersect at one point.

The microcalculator ''MK-97'' can work out the numbers entered in memory, perform only three operations: a) check whether the selected two numbers are equal; b) add the selected numbers; c) using the selected numbers $a$ and $b$, find the equation $x^2 +ax+b = 0$, and if there are no roots, display a message about this. The results of all actions are stored in memory. Initially, one number $x$ is stored in memory. How to use ''MK-97'' to find out whether is this number one?

Circles $S_1$ and $S_2$ intersect at points $M$ and $N$. Prove that if vertices $A$ and $ C$ of some rectangle $ABCD$ lie on the circle $S_1$, and the vertices $B$ and $D$ lie on the circle $S_2$, then the point of intersection of its diagonals lies on the line $MN$.

Natural numbers $m$ and $n$ are given. Prove that the number $2^n-1$ is divisible by the number $(2^m -1)^2$ if and only if the number $n$ is divisible by the number $m(2^m-1)$.

Given a cube with a side of $4$. Is it possible to completely cover $3$ of its faces, which have a common vertex, with sixteen rectangular paper strips measuring $1 \times3$?

Given a set of $100$ different numbers such that if each number in the set is replaced by the sum of the others, the same set will be obtained. Prove that the product of numbers in a set is positive.

In Mexico City, in order to limit traffic flow, for each private car is given one day a week on which it cannot go on the city streets. A wealthy family of 10 bribed the police, and for each car they are given 2 days, one of which the police chooses as a ''no travel'' day. What is the smallest number of cars a family needs to buy so that each family member can drive independently every day, if the approval of “no travel” days for cars occurs sequentially? original wordingВ городе Мехико в целях ограничения транспортного потока для каждой частной автомашины устанавливаются один деньв неделю, в который она не может выезжать на улицы города. Состоятельная семья из 10 человек подкупила полицию, и для каждой машины они называют 2 дня, один из которых полиция выбирает в качестве ''невыездного'' дня. Какое наименьшее количество машин нужно купить семье, чтобы каждый день каждый член семьи мог самостоятельно ездить, если утверждение ''невыездных'' дней для автомобилей идет последовательно?

Points $O_1$ and $O_2$ are the centers of the circumscribed and inscribed circles of an isosceles triangle $ABC$ ($AB = BC$). The circumcircles of triangles $ABC$ and $O_1O_2A$ intersect at points $A$ and $D$. Prove that line $BD$ is tangent to the circumcircle of the triangle $O_1O_2A$.

Prove that if $$\sqrt{x + a} +\sqrt{y+b}+\sqrt{z + c} =\sqrt{y + a} +\sqrt{z + b} +\sqrt{x + c} =\sqrt{z + a} +\sqrt{x+b}+\sqrt{y+c}$$for some $a, b, c, x, y, z$, then $x = y = z$ or $a = b = c$.

All vertices of triangle $ABC$ lie inside square $K$. Prove that if all of them are reflected symmetrically with respect to the point of intersection of the medians of triangle $ABC$, then at least one of the resulting three points will be inside $K$.

Let us denote by $S(m)$ the sum of the digits of the natural number $m$. Prove that there are infinitely many positive integers $n$ such that $$S(3^n) \ge S(3^{n+1}).$$

Members of the State Duma formed factions in such a way that for any two fractions $A $ and $B$ (not necessarily different), $\overline{A \cup B}$ is also faction ($\overline{C}$ denotes the set of all members of the Duma, not including in $C$). Prove that for any two factions $A$ and $B$, $A \cup B$ is also a faction.

Prove that if $1 < a < b < c$, then $$\log_a(\log_a b) + \log_b(\log_b c) + \log_c(\log_c a) > 0.$$

Are there convex $n$-gonal ($n \ge 4$) and triangular pyramids such that the four trihedral angles of the $n$-gonal pyramid are equal trihedral angles of a triangular pyramid? original wordingСуществуют ли выпуклая n-угольная (n>= 4) и треугольная пирамиды такие, что четыре трехгранных угла n-угольной пирамиды равны трехгранным углам треугольной пирамиды?

For which $a$, there is a function $f: R \to R$, different from a constant, such that $$f(a(x + y)) = f(x) + f(y) ?$$